Math 5125 Monday, September 12 September 12, Ungraded Homework Let H be a subgroup of the ﬁnite group G , and let p be a prime. Prove that two distinct Sylow p-subgroups of H cannot be contained in the same p-subgroup of G . Let P 1 , P 2 be Sylow p-subgroups of H , and suppose Q is a p-subgroup of G containing P 1 and P 2 . Since Q ∩ H is a p-group containing P 1 and P 1 is a Sylow p-subgroup, we must have Q ∩ H = P 1 . Similarly Q ∩ H = P 2 and we conclude that P 1 = P 2 , as required. Exercise 5.2.1(e) on page 165 Determine the number of nonisomorphic abelian groups of order 2704. First we write 2704 as a product of prime powers, namely 2 4 · 13 2 . To ﬁnd the number of abelian groups of order 16, we ﬁnd the number of partitions of 4. The partitions are (4), (3,1), (2,2), (2,1,1), (1,1,1,1) (so for example (3,1) corresponds to the group Z / 8 Z × Z / 2 Z ). Thus there are 5 abelian groups of order 16. Also there are 2 abelian groups of order 9. Therefore the total number of abelian groups of order 2704 is 5 · 2 = 10. Exercises 5.2.2(e) and 5.2.3(e) on page 165 List the elementary divisors and invariant

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